ON A NON-CORRECT PROBLEM FOR A BIHARMONIC EQUATION IN A SEMICIRCLE - Tolipov Nodirjon Isaqovich; Botirova Nasiba Djurabayevna (Modern American Journals, 2025)
This work investigates a conditionally correct problem for a biharmonic equation in a semicircular domain. The problem involves finding a function satisfying a biharmonic equation with mixed boundary conditions, including Dirichlet and Neumann-type constraints. It is demonstrated that the solution does not depend continuously on the input data, confirming the ill-posed nature of the problem. A stability estimate for the solution is derived under an a priori bound, and a family of regularizing operators is introduced to construct approximate solutions from noisy data. The effectiveness of the regularization method is analyzed, and an optimal parameter choice is discussed. An auxiliary problem is also formulated and reduced to a Fredholm integral equation of the first kind, which is addressed using Tikhonov regularization.
Keywords: Biharmonic equation, semicircle, ill-posed problem, conditional correctness, stability estimate, regularization method, Tikhonov regularization, Fredholm integral equation.